- Email: [email protected]
Tracking control and generalized projective synchronization of a class of hyperchaotic system with unknown parameter and disturbance
Commun Nonlinear Sci Numer Simulat 17 (2012) 405–413
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Tracking control and generalized projective synchronization of a class of hyperchaotic system with unknown parameter and disturbance Chunlai Li ⇑ College of Physics and Electronics, Hunan Institute of Science and Technology, Yueyang 414006, China
a r t i c l e
i n f o
Article history: Received 28 August 2010 Received in revised form 1 March 2011 Accepted 10 May 2011 Available online 18 May 2011 Keywords: Generalized projective synchronization Tracking control Unknown parameter Disturbance
a b s t r a c t In this paper, the tracking control and generalized projective synchronization of a class of hyperchaotic system with unknown parameter and disturbance are investigated. Based on the LaSalle’s invariant set theorem, a robust adaptive controller is contrived to acquire tracking control and generalized projective synchronization and parameter identification simultaneously. It is proved theoretically that the proposed scheme can allow us to drive the hyperchaotic system to any desired reference signals, including hyperchaotic signals, chaotic signals, periodic orbits or fixed value by the given scaling factor. The presented simulation results further demonstrate that the proposed method is effective and robust. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction The investigation of chaos control and synchronization have received a great deal of attention over the last decades because of the wide-scope potential applications in many disciplines such as secure communication, information science, biology, chemistry and engineering [1–6]. Among them, tracking control is the most commonly discussed problem in the domain of chaos control, which was introduced by Schwartz and Triandaf as a continuation method designed for experiments in the context of chaos control [7]. It can be explained that, for the arbitrary given reference signal, a controller is designed to cause the output of the chaotic system to follow the given reference signal asymptotically. Especially, if the reference signal is produced by the chaotic or hyperchaotic system, the tracking control evolves into synchronization. There are many studies on the tracking control in the literature, see in [8–10]. But it is a pity that all these methods about tracking control cannot make the variables of drive system track the given reference signal by a given scaling factor and cannot track different chaotic systems simultaneously. Nearly a decade research in chaos synchronization has focused on the concept of generalized projective synchronization which can expand the mode for encoding data and achieve communication rapidly [11–13]. Recently, Min [14] have combined tracking control with generalized projective synchronization ingeniously and formed a kind of program called tracking generalized projective synchronization which would have a strong anti-crack ability when used for secure communication. Subsequently, Li and Luo [15] introduced an accelerated scheme of tracking generalized projective synchronization for a fifth-order circuit’s hyperchaotic system. However, these methods mentioned above only concern some special dynamic systems, what’s more, these proposed techniques assume that the involved systems are free from unknown parameters and external perturbations. However, in practice we may not have this scenario, and have to take parameter uncertainty (or unknown) and external disturbances into account. The effect of these uncertainties will destroy the performance of synchronization or control and even break it. Therefore, it would be very instructive and significant to ⇑ Tel./fax: +86 730 8640052. E-mail address: [email protected] 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.05.017
406
C. Li / Commun Nonlinear Sci Numer Simulat 17 (2012) 405–413
study tracking control and generalized projective synchronization in such systems with unknown parameters and disturbances. Up till now, there is no more report on the program of tracking generalized projective synchronization. In this paper, for a class of chaotic or hyperchaotic systems by a unified mathematical expression, we propose a novel scheme for tracking control and generalized projective synchronization based on the LaSalle’s invariant set theorem. The proposed scheme is robust to the external disturbances and can identify the fully uncertain parameters simultaneously. So the proposed scheme is more essential and realistic in actual applications. Finally, simulation results are presented to demonstrate the effectiveness and robustness of the proposed method. And in addition, extensive numerical experiments show that the transient period of generalized projective synchronization and parameter identification is shorter with the increasing of control parameters L and j. 2. System model and problem descriptions In this paper, we consider a class hyperchaotic system which can be described by the following form,
x_ ¼ f ðxÞ þ FðxÞ#
ð1Þ T
n
nn
where x = (x1, x2, . . . , xn) is the state vector of the system, f(x) 2 R is the linear or nonlinear functions, F(x) 2 R , # = (#1, #2, . . . , #n)T are the parameters of the system, the number of the parameters is equal to the dimension of the system. The class of nonlinear dynamical systems include an extensive variety of hyperchaotic systems such as hyperchaotic Lorenz system, hyperchaotic Lü system, and so on. Let’s suppose that the parameters # are unknown and the hyperchaotic system is disturbed by the exotic perturbation n(t) 2 Rn which satisfy the bounded condition kn(t)k 6 k < 1 for all t. Then the controlled hyperchaotic system with unknown parameters and disturbances can be described by
x_ ¼ f ðxÞ þ FðxÞ# þ nðtÞ þ u
ð2Þ
n
where u 2 R is the controller to be designed, # are the unknown parameters, n(t) are the disturbances. Let r = (r1, r2, . . . , rn)T be the arbitrary given reference signal with first derivative. The synchronization error between system (2) and the reference signal is defined as e = x pr, where p = diag(p1, p2, . . . , pn) is called the scaling factor. Our aim is that, according to the designed controller, the all corresponding variables of system (2) follow the reference signal r proportionally. That is limt?1kek = limt?1kx prk = 0, where kk denotes a 2-norm in Rn. Then the dynamic equation of synchronization error can be expressed as follows:
e_ ¼ f ðxÞ þ FðxÞ# þ nðtÞ þ u pr_
ð3Þ
The proposed approach is a tool for achieving different types of synchronization. Namely, ifp = 1, identical synchronization is achieved; whereas for p = 1, anti-phase synchronization is obtained. And when jpij < 1 the range of the phase space amplifies; when jpij > 1 the range of the phase space shrinks. In addition, if the reference signal is periodic signal or fixed value, tracking control is obtained. 3. Design of adaptive control scheme
Theorem 1. For the drive system (2) and the arbitrary reference signal, if the controller u is designed as
^ u ¼ pr_ Le f ðxÞ FðxÞ#^ ke=kek
ð4Þ T
where L > 0 is the feedback strength and #^ ¼ ð#^1 ; #^2 ; . . . ; #^n Þ is the estimate of # which satisfy the following updated algorithm
_ #^ ¼ j½FðxÞT e
ð5Þ
^ complies with the following algorithm where j = diag(j1, j2, . . . , jn), j1, j2, . . . , jn > 0. And k
^_ ¼ kek k
ð6Þ
then all the corresponding variables of system (2) will approach the reference signal r ultimately according to the scaling factor, and all the unknow parameters # can be estimated by #^ asymptotically. ~ ¼kk ^ and the Lyapunov function of system (3) is constructed as ^ k Proof. Let #~ ¼ # #;
V¼
1 T 1 ~T ~ 1 ~T ~ e eþ # #þ k k 2 2j 2
Taking the time derivative of V along the generalized projective synchronization error and applying the renewal algorithm (5) and (6) yields
C. Li / Commun Nonlinear Sci Numer Simulat 17 (2012) 405–413
407
_~ j þ k ~_ ¼ ðf ðxÞ þ FðxÞ#ðtÞ þ nðtÞ þ u pr_ ÞT e #~T ð½FðxÞT eÞ k ~T k ~T kek V_ ¼ e_ T e þ #~T #= ~T kek 6 ðf ðxÞ þ FðxÞ#ðtÞ þ u pr_ ÞT e þ knðtÞk kek #~T ð½FðxÞT eÞ k ~T kek ¼ ðf ðxÞ þ FðxÞ#ðtÞ þ u pr_ ÞT e þ kkek ^ 6 ðf ðxÞ þ FðxÞ#ðtÞ þ u pr_ ÞT e þ kkek #~T ð½FðxÞT eÞ k #~T ð½FðxÞT eÞ ^ ^ ¼ ðf ðxÞ þ FðxÞ#ðtÞ þ pr_ Le f ðxÞ FðxÞ#^ ke=kek pr_ ÞT e þ kkek #~T ð½FðxÞT eÞ ^ T e #~T ð½FðxÞT eÞ ¼ LeT e 6 0 ¼ ðFðxÞ#ðtÞ Le FðxÞ#Þ ~ ¼ 0g is the largest invariant set which is contained in the set It is obvious that the set M ¼ fe ¼ 0; #~ ¼ 0; k E ¼ fejV_ ¼ 0; e 2 R4 g. So, according to the LaSalle’s invariant set theorem [17], all the corresponding variables of system (2) will follow the reference signal r asymptotically according to the scaling factor, and the parameters # can be estimated by #^ ultimately. h
4. Numerical simulations In this section, three numerical examples are presented to demonstrate and verify the effectiveness and the robustness of the proposed approach. We take a new hyperchaotic system as the drive system here, which can be described by Wei et al. [16]
8 x_ 1 > > > < x_ 2 > x_ 3 > > : x_ 4
¼ aðx2 x1 Þ ¼ cx1 x1 x3 þ x4
ð7Þ
¼ bx3 þ x1 x2 ¼ dx1
The system will exhibit a hyperchaotic behavior under the following conditions: a = 35, b = 3, c = 35, d = 8. And the two positive Lyapunov exponents is k1 = 0.2788, k2 = 0.1470. Now, we present the dynamics of system (7) in the following form,
x_ ¼ f ðxÞ þ FðxÞ#
ð8Þ
where
2
x1
3
6x 7 6 27 x ¼ 6 7; 4 x3 5
3
2
6 x x þ x 7 47 6 1 3 f ðxÞ ¼ 6 7; 5 4 x1 x2
6 6 FðxÞ ¼ 6 4
2
x4
0
0
3
x2 x1
0
0
0
0
x1
0
0
0
x3
0 7 7 7; 0 5
0
0
0
x1
2 3 a 6c 7 6 7 #¼6 7 4b5 d
Then the controlled hyperchaotic system with unknown parameters and disturbances can be described in the following form,
x_ ¼ f ðxÞ þ FðxÞ# þ nðtÞ þ u
ð9Þ
For convenience and comparison, in all the process of simulation, the fourth-order Runge–Kutta method is used with time step 0.001, the initial states of the new hyperchaotic system are taken as x1(0) = 0.2, x2(0) = 0.5, x3(0) = 0.5, x4(0) = 0.8, the ^ ¼ 5:4; ^c ¼ 0:2; d ^ ¼ 0:8; ^ ¼ 5:7; b initial values of the estimate of the unknown parameters are chosen as a L ¼ 5; j1 ¼ 5; j2 ¼ 5; j3 ¼ 1:5; j4 ¼ 4:5, and the exotic disturbance f(t) = [n1, n2, n3, n4]T, n1, n2, n3, n4 are stochastic number of (0, 1). 4.1. Generalized projective synchronization of identical hyperchaotic systems First, we take the new hyperchaotic system as the driven system realizing generalized projective synchronization. The driven system is described as
8 y_ 1 > > > < y_ 2 > > y_ 3 > : y_ 4
¼ aðy2 y1 Þ ¼ cy1 y1 y3 þ y4 ¼ by3 þ y1 y2
ð10Þ
¼ dy1
And we take the all variables of driven system (10) as the reference signals, i.e. r = (y1, y2, y3, y4)T. The initial conditions of system (10) are y1(0) = 1, y2(0) = 0.1, y3(0) = 0.01, y4(0) = 1. And let p = diag(2, 2, 0.5, 1). The simulation results are shown in Fig. 1. Fig. 1(a) represent the time evolution of variable x and reference signal r, the dot line is the variable and the solid line is the corresponding reference signal r. Fig. 1(b) displays the time evolution of synchronization errors. And the time evolutions of adaptive parameters are exhibited Fig. 1(c). As we can see that the systems (9) and (10) have achieved generalized projective synchronization ultimately, and all the unknow parameters # are estimated by #^ asymptotically.
408
C. Li / Commun Nonlinear Sci Numer Simulat 17 (2012) 405–413
Fig. 1. Simulation results of generalized projective synchronization between systems (9) and (10): (a) time evolution of x and r; (b) synchronization errors e1, e2, e3 and e4; (c) time evolution of the estimated parameters.
4.2. Tracking two different chaotic systems In this section, two different chaotic systems, Van der Pol–Duffing oscillator and Bonhoeffer–van der pol oscillator, will be choosed as the reference signal to illustrate the validity of the proposed approach.
C. Li / Commun Nonlinear Sci Numer Simulat 17 (2012) 405–413
409
Fig. 2. Simulation results of generalized projective synchronization between system (9), (11) and (12): (a) synchronized attractors of systems (9), (11) and (12); (b) synchronization errors e1, e2, e3 and e4; (c) time evolution of estimated parameters.
The Van der Pol–Duffing oscillator can be given as [18]
y_ 1 ¼ y2 y_ 2 ¼ lð1 y21 Þy2 þ ay1 by31 þ f cosðxtÞ
when the parameters are set equal to l = 0.1, a = b = 1, x = 1, f = 3, the system is experiencing chaotic behavior.
ð11Þ
410
C. Li / Commun Nonlinear Sci Numer Simulat 17 (2012) 405–413
Fig. 3. Simulation results of the hyperchaotic system (9) tracking the periodic signal and fixed value: (a) time evolution of x and r; (b) synchronization errors e1, e2, e3 and e4; (c) time evolution of the estimated parameters.
And the Bonhoeffer–van der pol oscillator can be described by [19]:
(
y_ 3 ¼ y3 y33 =3 y4 þ f cos t y_ 4 ¼ cðy3 þ a by4 Þ
where the parameters are set equal to a = 0.7, b = 0.8, c = 0.1, f = 0.74 with which the system will behave chaotically.
ð12Þ
C. Li / Commun Nonlinear Sci Numer Simulat 17 (2012) 405–413
411
Fig. 4. Simulation results of generalized projective synchronization between system (9) and (10) with L = 2, j1 = 2, j2 = 2, j3 = 1.5, j4 = 2.5: (a) synchronization errors e1, e2, e3 and e4; (b) parameter identification errors.
Now, we choose Van der Pol–Duffing oscillator and Bonhoeffer–van der pol oscillator as the reference signal, that is, r = (y1, y2, y3, y4)T, and let p = diag(2, 2, 0.5, 0.5). The initial conditions of systems (11) and (12) are set to y1(0) = 1, y2(0) = 0.1, y3(0) = 0.01, y4(0) = 1. The simulation results are shown in Fig. 2. Fig. 2(a) shows the synchronized attractors of systems (9), (11) and (12), the dot line represent the variable and the solid line is the corresponding reference signal. Fig. 2(b) are the synchronization errors. And the time evolutions of adaptive parameters are exhibited Fig. 2(c). Similarly, we know the system (9), (11) and (12) have achieved generalized projective synchronization ultimately, and the unknow parameters # are identified by #^ asymptotically.
4.3. Tracking the periodic signal and fixed value Now, we choose the sinusoidal signal and fixed value as the reference signal, that is, r = (cos2 t, sin3 t + cos2 t, sin3t, 2)T, and p = diag(0.5, 1, 2, 1). The simulation results are shown in Fig. 3. Fig. 3(a) represent the time evolution of variable x and reference signal r, the dot line is the variable and the solid line is the corresponding reference signal. Fig. 3(b) shows the time evolution of synchronization errors. Fig. 3(c) exhibit the time histories of adaptive parameters. As we can see that the system (9) has tracked the periodic signal and fixed value ultimately according to the scaling factor, and all the unknow parameters # are estimated by #^ asymptotically. In addition, extensive numerical experiments show that the transient period is shorter when bigger feedback strength L and j are used. As a example, the generalized projective synchronization of identical hyperchaotic systems is considered here. We first set the parameters L = 2, j1 = 2, j2 = 2, j3 = 1.5, j4 = 2.5, and the simulation results are shown in Fig. 4. Fig. 4(a) show the synchronized errors, Fig. 4(b) are the parameter identification errors, the transient time of synchronization and parameter identification is about 1.8 s. And Fig. 5 show the simulation results with L = 8, j1 = 7, j2 = 7, j3 = 6.5, j4 = 6.5, and the transient time of synchronization and parameter identification is about 0.9 s.
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C. Li / Commun Nonlinear Sci Numer Simulat 17 (2012) 405–413
Fig. 5. Simulation results of generalized projective synchronization between systems (9) and (10) with L = 8, j1 = 7, j2 = 7, j3 = 6.5, j4 = 6.5: (a) synchronization errors e1, e2, e3 and e4; (b) parameter identification errors.
5. Conclusion In this paper, an adaptive controller of tracking control and generalized projective synchronization for a class of hyperchaotic system has been designed, which can allow us to drive the hyperchaotic system to any given reference signal according to the scaling factor freely and can identify the unknown parameter of drive system simultaneously. The proposed controller is robust and effective. Simulation results are presented to demonstrate the validity of the proposed method. References [1] Wang GR, Yu XL, Chen SG. Chaotic control. Synchronization and utilizing. Beijing: National Defence Industry Press; 2001. [2] Ngueuteu GSM, Yamapi R, Woafo P. Effects of higher nonlinearity on the dynamics and synchronization of two coupled electromechanical devices. Commun Nonlinear Sci Numer Simul 2008;13:1213–40. [3] Cao ZJ, Li PF, Zhang H, Xie FG, Hu G. Turbulence control with local pacing and its implication in cardiac defibrillation. Chaos 2007;17:0151071–79. [4] Guo WL, Chen SH, Zhou H. A simple adaptive-feedback controller for chaos synchronization. Chaos Soliton Fract 2009;39:316–21. [5] Holstein-Rathlou NH, Yip KP, Sosnovtseva OV, Mosekildem E. Synchronization phenomena in nephron–nephron interaction. Chaos 2001;11:417–26. [6] Feki M. An adaptive chaos synchronization scheme applied to secure communication. Chaos Soliton Fract 2003;18:141–8. [7] Schwartz IB, Triandaf I. Tracking unstable orbits in experiments. Phys Rev A 1992;46:7439–44. [8] Wang XY, Wu XJ. Tracking control and synchronization of four-dimension hyperchaotic Rossler system. Chaos 2006;16:033121–8. [9] Li J, Lin H, Li N. Chaotic synchronization with diverse structures based on tracking control. Acta Phys Sin 2006;55:3992–7. [10] Li Z, Chen G, Shi S, Han C. Robust adaptive tracking control for a class of uncertain chaotic systems. Phys Lett A 2003;310:40–3. [11] Meng J, Wang XY. Generalized projective synchronization of a class of delayed neural networks. Modern Phys Lett B 2008;22:181–90. [12] Li CP, Yan JP. Generalized projective synchronization of chaos: the cascade synchronization approach. Chaos Soliton Fract 2006;30:140–6. [13] Jiang DP, Luo XS, Wang BH, Fang JQ, Jiang PQ. Study on proportional synchronization of hyperchaotic circuit system. Commun Theor Phys 2005;43:671–6. [14] Min FH, Wang ZQ. Generalized projective synchronization and tracking control of complex dynamous systems. Acta Phys Sin 2008;57:0031–6. [15] Li CL, Luo XS. Tracking control and projective synchronization in the fifth-order hyperchaotic circuit system based on accelerated factor. Acta Phys Sin 2009;58:3759–64. [16] Wei DQ, Zhang B, Qiu DY, Luo XS. Adaptive controlling chaos in permanent magnet synchronous motor based on the LaSalle theory. Acta Phys Sin 2009;58:6026–9.
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[17] Liu MH, Feng JC. A new hyperchaotic system. Acta Phys Sin 2009;58:4457–62. [18] Yuuichi Susuki, Yokoi Yoshihiko, Takashi Hikihara. Energy-based analysis of frequency entrainment described by van der Pol and phase-locked loop equations. Chaos 2007;17:0231081–88. [19] Zhang Y, Xu W, Fang T. Stochastic Hopf bifurcation and chaos of stochastic Bonhoeffer–van der Pol system via Chebyshev polynomial approximation. Appl Math Comput 2007;190:1225–36.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Tracking control and generalized projective synchronization of a class of hyperchaotic system with unknown parameter and disturbance Chunlai Li ⇑ College of Physics and Electronics, Hunan Institute of Science and Technology, Yueyang 414006, China
a r t i c l e
i n f o
Article history: Received 28 August 2010 Received in revised form 1 March 2011 Accepted 10 May 2011 Available online 18 May 2011 Keywords: Generalized projective synchronization Tracking control Unknown parameter Disturbance
a b s t r a c t In this paper, the tracking control and generalized projective synchronization of a class of hyperchaotic system with unknown parameter and disturbance are investigated. Based on the LaSalle’s invariant set theorem, a robust adaptive controller is contrived to acquire tracking control and generalized projective synchronization and parameter identification simultaneously. It is proved theoretically that the proposed scheme can allow us to drive the hyperchaotic system to any desired reference signals, including hyperchaotic signals, chaotic signals, periodic orbits or fixed value by the given scaling factor. The presented simulation results further demonstrate that the proposed method is effective and robust. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction The investigation of chaos control and synchronization have received a great deal of attention over the last decades because of the wide-scope potential applications in many disciplines such as secure communication, information science, biology, chemistry and engineering [1–6]. Among them, tracking control is the most commonly discussed problem in the domain of chaos control, which was introduced by Schwartz and Triandaf as a continuation method designed for experiments in the context of chaos control [7]. It can be explained that, for the arbitrary given reference signal, a controller is designed to cause the output of the chaotic system to follow the given reference signal asymptotically. Especially, if the reference signal is produced by the chaotic or hyperchaotic system, the tracking control evolves into synchronization. There are many studies on the tracking control in the literature, see in [8–10]. But it is a pity that all these methods about tracking control cannot make the variables of drive system track the given reference signal by a given scaling factor and cannot track different chaotic systems simultaneously. Nearly a decade research in chaos synchronization has focused on the concept of generalized projective synchronization which can expand the mode for encoding data and achieve communication rapidly [11–13]. Recently, Min [14] have combined tracking control with generalized projective synchronization ingeniously and formed a kind of program called tracking generalized projective synchronization which would have a strong anti-crack ability when used for secure communication. Subsequently, Li and Luo [15] introduced an accelerated scheme of tracking generalized projective synchronization for a fifth-order circuit’s hyperchaotic system. However, these methods mentioned above only concern some special dynamic systems, what’s more, these proposed techniques assume that the involved systems are free from unknown parameters and external perturbations. However, in practice we may not have this scenario, and have to take parameter uncertainty (or unknown) and external disturbances into account. The effect of these uncertainties will destroy the performance of synchronization or control and even break it. Therefore, it would be very instructive and significant to ⇑ Tel./fax: +86 730 8640052. E-mail address: [email protected] 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.05.017
406
C. Li / Commun Nonlinear Sci Numer Simulat 17 (2012) 405–413
study tracking control and generalized projective synchronization in such systems with unknown parameters and disturbances. Up till now, there is no more report on the program of tracking generalized projective synchronization. In this paper, for a class of chaotic or hyperchaotic systems by a unified mathematical expression, we propose a novel scheme for tracking control and generalized projective synchronization based on the LaSalle’s invariant set theorem. The proposed scheme is robust to the external disturbances and can identify the fully uncertain parameters simultaneously. So the proposed scheme is more essential and realistic in actual applications. Finally, simulation results are presented to demonstrate the effectiveness and robustness of the proposed method. And in addition, extensive numerical experiments show that the transient period of generalized projective synchronization and parameter identification is shorter with the increasing of control parameters L and j. 2. System model and problem descriptions In this paper, we consider a class hyperchaotic system which can be described by the following form,
x_ ¼ f ðxÞ þ FðxÞ#
ð1Þ T
n
nn
where x = (x1, x2, . . . , xn) is the state vector of the system, f(x) 2 R is the linear or nonlinear functions, F(x) 2 R , # = (#1, #2, . . . , #n)T are the parameters of the system, the number of the parameters is equal to the dimension of the system. The class of nonlinear dynamical systems include an extensive variety of hyperchaotic systems such as hyperchaotic Lorenz system, hyperchaotic Lü system, and so on. Let’s suppose that the parameters # are unknown and the hyperchaotic system is disturbed by the exotic perturbation n(t) 2 Rn which satisfy the bounded condition kn(t)k 6 k < 1 for all t. Then the controlled hyperchaotic system with unknown parameters and disturbances can be described by
x_ ¼ f ðxÞ þ FðxÞ# þ nðtÞ þ u
ð2Þ
n
where u 2 R is the controller to be designed, # are the unknown parameters, n(t) are the disturbances. Let r = (r1, r2, . . . , rn)T be the arbitrary given reference signal with first derivative. The synchronization error between system (2) and the reference signal is defined as e = x pr, where p = diag(p1, p2, . . . , pn) is called the scaling factor. Our aim is that, according to the designed controller, the all corresponding variables of system (2) follow the reference signal r proportionally. That is limt?1kek = limt?1kx prk = 0, where kk denotes a 2-norm in Rn. Then the dynamic equation of synchronization error can be expressed as follows:
e_ ¼ f ðxÞ þ FðxÞ# þ nðtÞ þ u pr_
ð3Þ
The proposed approach is a tool for achieving different types of synchronization. Namely, ifp = 1, identical synchronization is achieved; whereas for p = 1, anti-phase synchronization is obtained. And when jpij < 1 the range of the phase space amplifies; when jpij > 1 the range of the phase space shrinks. In addition, if the reference signal is periodic signal or fixed value, tracking control is obtained. 3. Design of adaptive control scheme
Theorem 1. For the drive system (2) and the arbitrary reference signal, if the controller u is designed as
^ u ¼ pr_ Le f ðxÞ FðxÞ#^ ke=kek
ð4Þ T
where L > 0 is the feedback strength and #^ ¼ ð#^1 ; #^2 ; . . . ; #^n Þ is the estimate of # which satisfy the following updated algorithm
_ #^ ¼ j½FðxÞT e
ð5Þ
^ complies with the following algorithm where j = diag(j1, j2, . . . , jn), j1, j2, . . . , jn > 0. And k
^_ ¼ kek k
ð6Þ
then all the corresponding variables of system (2) will approach the reference signal r ultimately according to the scaling factor, and all the unknow parameters # can be estimated by #^ asymptotically. ~ ¼kk ^ and the Lyapunov function of system (3) is constructed as ^ k Proof. Let #~ ¼ # #;
V¼
1 T 1 ~T ~ 1 ~T ~ e eþ # #þ k k 2 2j 2
Taking the time derivative of V along the generalized projective synchronization error and applying the renewal algorithm (5) and (6) yields
C. Li / Commun Nonlinear Sci Numer Simulat 17 (2012) 405–413
407
_~ j þ k ~_ ¼ ðf ðxÞ þ FðxÞ#ðtÞ þ nðtÞ þ u pr_ ÞT e #~T ð½FðxÞT eÞ k ~T k ~T kek V_ ¼ e_ T e þ #~T #= ~T kek 6 ðf ðxÞ þ FðxÞ#ðtÞ þ u pr_ ÞT e þ knðtÞk kek #~T ð½FðxÞT eÞ k ~T kek ¼ ðf ðxÞ þ FðxÞ#ðtÞ þ u pr_ ÞT e þ kkek ^ 6 ðf ðxÞ þ FðxÞ#ðtÞ þ u pr_ ÞT e þ kkek #~T ð½FðxÞT eÞ k #~T ð½FðxÞT eÞ ^ ^ ¼ ðf ðxÞ þ FðxÞ#ðtÞ þ pr_ Le f ðxÞ FðxÞ#^ ke=kek pr_ ÞT e þ kkek #~T ð½FðxÞT eÞ ^ T e #~T ð½FðxÞT eÞ ¼ LeT e 6 0 ¼ ðFðxÞ#ðtÞ Le FðxÞ#Þ ~ ¼ 0g is the largest invariant set which is contained in the set It is obvious that the set M ¼ fe ¼ 0; #~ ¼ 0; k E ¼ fejV_ ¼ 0; e 2 R4 g. So, according to the LaSalle’s invariant set theorem [17], all the corresponding variables of system (2) will follow the reference signal r asymptotically according to the scaling factor, and the parameters # can be estimated by #^ ultimately. h
4. Numerical simulations In this section, three numerical examples are presented to demonstrate and verify the effectiveness and the robustness of the proposed approach. We take a new hyperchaotic system as the drive system here, which can be described by Wei et al. [16]
8 x_ 1 > > > < x_ 2 > x_ 3 > > : x_ 4
¼ aðx2 x1 Þ ¼ cx1 x1 x3 þ x4
ð7Þ
¼ bx3 þ x1 x2 ¼ dx1
The system will exhibit a hyperchaotic behavior under the following conditions: a = 35, b = 3, c = 35, d = 8. And the two positive Lyapunov exponents is k1 = 0.2788, k2 = 0.1470. Now, we present the dynamics of system (7) in the following form,
x_ ¼ f ðxÞ þ FðxÞ#
ð8Þ
where
2
x1
3
6x 7 6 27 x ¼ 6 7; 4 x3 5
3
2
6 x x þ x 7 47 6 1 3 f ðxÞ ¼ 6 7; 5 4 x1 x2
6 6 FðxÞ ¼ 6 4
2
x4
0
0
3
x2 x1
0
0
0
0
x1
0
0
0
x3
0 7 7 7; 0 5
0
0
0
x1
2 3 a 6c 7 6 7 #¼6 7 4b5 d
Then the controlled hyperchaotic system with unknown parameters and disturbances can be described in the following form,
x_ ¼ f ðxÞ þ FðxÞ# þ nðtÞ þ u
ð9Þ
For convenience and comparison, in all the process of simulation, the fourth-order Runge–Kutta method is used with time step 0.001, the initial states of the new hyperchaotic system are taken as x1(0) = 0.2, x2(0) = 0.5, x3(0) = 0.5, x4(0) = 0.8, the ^ ¼ 5:4; ^c ¼ 0:2; d ^ ¼ 0:8; ^ ¼ 5:7; b initial values of the estimate of the unknown parameters are chosen as a L ¼ 5; j1 ¼ 5; j2 ¼ 5; j3 ¼ 1:5; j4 ¼ 4:5, and the exotic disturbance f(t) = [n1, n2, n3, n4]T, n1, n2, n3, n4 are stochastic number of (0, 1). 4.1. Generalized projective synchronization of identical hyperchaotic systems First, we take the new hyperchaotic system as the driven system realizing generalized projective synchronization. The driven system is described as
8 y_ 1 > > > < y_ 2 > > y_ 3 > : y_ 4
¼ aðy2 y1 Þ ¼ cy1 y1 y3 þ y4 ¼ by3 þ y1 y2
ð10Þ
¼ dy1
And we take the all variables of driven system (10) as the reference signals, i.e. r = (y1, y2, y3, y4)T. The initial conditions of system (10) are y1(0) = 1, y2(0) = 0.1, y3(0) = 0.01, y4(0) = 1. And let p = diag(2, 2, 0.5, 1). The simulation results are shown in Fig. 1. Fig. 1(a) represent the time evolution of variable x and reference signal r, the dot line is the variable and the solid line is the corresponding reference signal r. Fig. 1(b) displays the time evolution of synchronization errors. And the time evolutions of adaptive parameters are exhibited Fig. 1(c). As we can see that the systems (9) and (10) have achieved generalized projective synchronization ultimately, and all the unknow parameters # are estimated by #^ asymptotically.
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Fig. 1. Simulation results of generalized projective synchronization between systems (9) and (10): (a) time evolution of x and r; (b) synchronization errors e1, e2, e3 and e4; (c) time evolution of the estimated parameters.
4.2. Tracking two different chaotic systems In this section, two different chaotic systems, Van der Pol–Duffing oscillator and Bonhoeffer–van der pol oscillator, will be choosed as the reference signal to illustrate the validity of the proposed approach.
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Fig. 2. Simulation results of generalized projective synchronization between system (9), (11) and (12): (a) synchronized attractors of systems (9), (11) and (12); (b) synchronization errors e1, e2, e3 and e4; (c) time evolution of estimated parameters.
The Van der Pol–Duffing oscillator can be given as [18]
y_ 1 ¼ y2 y_ 2 ¼ lð1 y21 Þy2 þ ay1 by31 þ f cosðxtÞ
when the parameters are set equal to l = 0.1, a = b = 1, x = 1, f = 3, the system is experiencing chaotic behavior.
ð11Þ
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Fig. 3. Simulation results of the hyperchaotic system (9) tracking the periodic signal and fixed value: (a) time evolution of x and r; (b) synchronization errors e1, e2, e3 and e4; (c) time evolution of the estimated parameters.
And the Bonhoeffer–van der pol oscillator can be described by [19]:
(
y_ 3 ¼ y3 y33 =3 y4 þ f cos t y_ 4 ¼ cðy3 þ a by4 Þ
where the parameters are set equal to a = 0.7, b = 0.8, c = 0.1, f = 0.74 with which the system will behave chaotically.
ð12Þ
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Fig. 4. Simulation results of generalized projective synchronization between system (9) and (10) with L = 2, j1 = 2, j2 = 2, j3 = 1.5, j4 = 2.5: (a) synchronization errors e1, e2, e3 and e4; (b) parameter identification errors.
Now, we choose Van der Pol–Duffing oscillator and Bonhoeffer–van der pol oscillator as the reference signal, that is, r = (y1, y2, y3, y4)T, and let p = diag(2, 2, 0.5, 0.5). The initial conditions of systems (11) and (12) are set to y1(0) = 1, y2(0) = 0.1, y3(0) = 0.01, y4(0) = 1. The simulation results are shown in Fig. 2. Fig. 2(a) shows the synchronized attractors of systems (9), (11) and (12), the dot line represent the variable and the solid line is the corresponding reference signal. Fig. 2(b) are the synchronization errors. And the time evolutions of adaptive parameters are exhibited Fig. 2(c). Similarly, we know the system (9), (11) and (12) have achieved generalized projective synchronization ultimately, and the unknow parameters # are identified by #^ asymptotically.
4.3. Tracking the periodic signal and fixed value Now, we choose the sinusoidal signal and fixed value as the reference signal, that is, r = (cos2 t, sin3 t + cos2 t, sin3t, 2)T, and p = diag(0.5, 1, 2, 1). The simulation results are shown in Fig. 3. Fig. 3(a) represent the time evolution of variable x and reference signal r, the dot line is the variable and the solid line is the corresponding reference signal. Fig. 3(b) shows the time evolution of synchronization errors. Fig. 3(c) exhibit the time histories of adaptive parameters. As we can see that the system (9) has tracked the periodic signal and fixed value ultimately according to the scaling factor, and all the unknow parameters # are estimated by #^ asymptotically. In addition, extensive numerical experiments show that the transient period is shorter when bigger feedback strength L and j are used. As a example, the generalized projective synchronization of identical hyperchaotic systems is considered here. We first set the parameters L = 2, j1 = 2, j2 = 2, j3 = 1.5, j4 = 2.5, and the simulation results are shown in Fig. 4. Fig. 4(a) show the synchronized errors, Fig. 4(b) are the parameter identification errors, the transient time of synchronization and parameter identification is about 1.8 s. And Fig. 5 show the simulation results with L = 8, j1 = 7, j2 = 7, j3 = 6.5, j4 = 6.5, and the transient time of synchronization and parameter identification is about 0.9 s.
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Fig. 5. Simulation results of generalized projective synchronization between systems (9) and (10) with L = 8, j1 = 7, j2 = 7, j3 = 6.5, j4 = 6.5: (a) synchronization errors e1, e2, e3 and e4; (b) parameter identification errors.
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